#9034385

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4926 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2,4,6)(3,5,7,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,21)(20,23)(22,24)(25,26,29,31)(27,30,33,39)(28,35,37,32)(34,40,36,38);(3,7)(5,8)(10,15)(12,21)(16,23)(18,22)(25,27)(26,30)(28,34)(29,33)(31,39)(32,38)(35,40)(36,37);(1,2)(3,5)(4,6)(7,8)(12,21)(13,24)(17,19)(18,22)(26,31)(27,33)(32,35)(34,36);(9,11)(10,16)(12,22)(13,19)(14,20)(15,23)(17,24)(18,21)(25,28)(26,32)(27,34)(29,37)(30,38)(31,35)(33,36)(39,40);(1,3)(2,5)(4,7)(6,8)(9,12,14,21)(10,17,15,19)(11,18,20,22)(13,23,24,16)(25,27)(26,30)(28,36)(29,33)(31,39)(32,40)(34,37)(35,38);(1,2)(3,5)(4,6)(7,8)(9,13)(10,18)(11,19)(12,23)(14,24)(15,22)(16,21)(17,20)(25,26)(27,30)(28,32)(29,31)(33,39)(34,38)(35,37)(36,40)

Polycyclic presentation:

2^9|a^2=h,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8941581);